Proving the left lifting property for a map

Question

I want to prove the left lifting property for the inclusion map of the sphere into the disk for any fibration $q:X \rightarrow Y$, where $q$ is a weak equivalence.

I don't know how to draw a square diagram here ( my latex code isn't working for some reason) .... So please bear with me :)

So, what I want to show that for any maps $ f_0: S^n \rightarrow X$ and any map $f_1: D^{n+1} \rightarrow Y$ such that $ f_1 \circ i=q \circ f_0$;

I can find a map $f: D^{n+1} \rightarrow X$ such that $ f \circ i = f_0$ and $ q \circ f = f_1$.

Any help finding this map?

Thanks!

Answer 1

Write $D^{n+1}$ as a quotient of $S^{n} \times [0,1] $. You can then apply fibration property to get a map $S^{n} \times [0,1] \rightarrow X$ such that $S^{n} \times \{0\}$ and $S^{n} \times \{1\}$ both map to a single fiber. Because of the weak equivalence, $\pi_n$ of each fiber is $0$ and so you can contract each of these to get a map $D^{n+1} \rightarrow X$.